PlanetPhysics/Overview of Algebraic Topology

An Overview of Algebraic Topology topics

Introduction

Algebraic topology (AT) utilizes [[../CoIntersections/|algebraic]] approaches to solve [[../CoIntersections/|topological]] problems, such as the [[../TrivialGroupoid/|classification]] of surfaces, proving [[../DualityAndTriality/|duality]] [[../Formula/|theorems]] for manifolds and approximation theorems for topological spaces. A central problem in algebraic topology is to find algebraic invariants of topological spaces, which is usually carried out by means of [[../ThinEquivalence/|homotopy]], homology and [[../CohomologyTheoryOnCWComplexes/|cohomology groups]]. There are close connections between algebraic topology, Algebraic Geometry (AG)

1. [[../CubicalHigherHomotopyGroupoid/|homotopy theory]] and [[../Pushout/|fundamental groups]] #Topology and [[../GroupoidHomomorphism2/|groupoids]]; van Kampen theorem
2. Homology and cohomology theories
3. Duality
4. [[../CategoricalOntology/|category theory applications]] in algebraic topology
5. [[../IndexOfCategories/|indexes of category]], [[../TrivialGroupoid/|functors]] and [[../VariableCategory2/|natural transformations]]
6. Grothendieck's Descent theory
7. `[[../IsomorphismClass/|Anabelian Geometry]]' #Categorical Galois theory
8. [[../HigherDimensionalAlgebra2/|higher dimensional algebra]] ([[../2Groupoid2/|HDA]])
9. Quantum algebraic topology ([[../NonNewtonian2/|QAT]])
10. Non-Abelian Quantum Algebraic Topology

($http://aux.planetphysics.org/files/lec/61/ANAQAT20c.pdf$)

1. Quantum Geometry
2. Non-Abelian algebraic topology (NAAT)

Homotopy theory and fundamental groups

1. Homotopy
2. [[../ModuleAlgebraic/|fundamental group]] of a space
3. Fundamental theorems
4. [[../VanKampenTheorems/|Van Kampen theorem]] #Whitehead [[../TrivialGroupoid/|groups]], torsion and towers
5. Postnikov towers

Topology and Groupoids

1. Topology definition, axioms and basic [[../PreciseIdea/|concepts]] #[[../CubicalHigherHomotopyGroupoid/|fundamental groupoid]] #[[../GroupoidHomomorphism2/|topological groupoid]] #van Kampen theorem for groupoids
2. Groupoid [[../Pushout/|pushout]] theorem
3. [[../ThinEquivalence/|double groupoids]] and [[../CubicalHigherHomotopyGroupoid/|crossed modules]]
4. new4

Homology theory

1. Homology group
2. Homology sequence
3. Homology complex
4. new4

Cohomology theory

1. [[../CubicalHigherHomotopyGroupoid/|cohomology group]] #Cohomology sequence
2. DeRham cohomology
3. new4

Non-Abelian Algebraic Topology

1. [[../SingularComplexOfASpace/|crossed complexes]] #[[../RModule/|modules]] #[[../SingularComplexOfASpace/|omega-groupoids]] #double groupoids
2. Higher Homotopy van Kampen Theorems